Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

In the present chapter we consider the well-posedness of an abstract boundary-value problem for differential equations of elliptic type

$$- \upsilon ''\left( t \right) + A\upsilon \left( t \right) = f\left( t \right)\left( {0 \leqslant t \leqslant T} \right),\upsilon \left( 0 \right) = {{\upsilon }_{0}},\upsilon \left( T \right) = {{\upsilon }_{T}}$$

in an arbitrary Banach spaceEwith the positive operator A. The high order of accuracy two-step difference schemes generated by an exact difference scheme or by the Taylor decomposition on three points for the numerical solutions of this problem are presented. The well-posedness of these difference schemes in various Banach spaces are studied. The stability and coercive stability estimates in Holder norms for solutions of the high order of accuracy difference schemes of the mixed type boundary-value problems for elliptic equations are obtained.

Partial Differential Equations of Elliptic Type

https://bura.brunel.ac.uk/bitstream/2438/5913/2/Mikhailov-JMAA2011-6.pdf

Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains

Cosserat models of cancellous bone are constructed, relying on micromechanical approaches in order to investigate microstructure-related scale effects on the macroscopic properties of bone. The derivation of the effective mechanical properties of cancellous bone considered as a cellular solid modeled as two-dimensional lattices of articulated beams is presently investigated. The cell walls of the bone microstructure are modeled as Timoshenko thick beams. The asymptotic homogenization technique is involved to get closed form expressions of the equivalent properties versus the geometrical and mechanical microparameters, accounting for the effects of bending, axial, and transverse shear deformations. Considering lattice microrotations as additional degrees of freedom at both the microscopic and macroscopic scales, an anisotropic micropolar equivalent continuum model is constructed, the effective mechanical properties of which are identified. The effective elastic moduli of various periodic cell structures are computed in situations of low and high effective densities to assess the impact of the transverse shear deformation. The stress distribution in a cracked bone sample is computed based on the effective micropolar model, highlighting the regularizing effect of the Cosserat continuum in comparison to a classical elasticity continuum model.

A micropolar anisotropic constitutive model of cancellous bone from discrete homogenization.

A 3D anisotropic micropolar continuum model of vertebral trabecular bone is presently developed accounting for the influence of microstructure-related scale effects on the macroscopic effective properties. Vertebral trabecular bone is modeled as a cellular material with an idealized periodic structure made of open 3D cells. The micromechanical approach relies on the discrete homogenization technique considering lattice microrotations as additional degrees of freedom at the microscale. The effective elastic properties of 3D lattices made of articulated beams taking into account axial, transverse shearing, flexural, and torsional deformations of the cell struts are derived as closed form expressions of the geometrical and mechanical microparameters. The scaling laws of the effective moduli versus density are determined in situations of low and high effective densities to assess the impact of the transverse shear deformation. The classical and micropolar effective moduli and the internal flexural and torsional lengths are identified versus the same microparameters. A finite element model of the local architecture of the trabeculae gives values of the effective moduli that are in satisfactory agreement with the homogenized moduli.

A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure

In this paper, we solve fundamental boundary value problems in a theory of antiplane elasticity which includes the effects of material microstructure. Using the real boundary integral equation method, we reduce the fundamental problems to systems of singular integral equations and construct exact solutions in the form of integral potentials.

Antiplane shear deformations in a linear theory of elasticity with microstructure

Boundary element analysis of stress distribution around a crack in plane micropolar elasticity

https://uwaterloo.ca/centre-advanced-materials-joining/sites/ca.centre-advanced-materials-joining/files/uploads/files/7_e_astroshchenko_2009.pdf

Stress intensity factor for an embedded elliptical crack under arbitrary normal loading

In this paper we formulate interior Dirichlet and Neumann boundary value problems of plane Cosserat elasticity in Sobolev spaces, show that these problems are well-posed and find the corresponding weak solutions in the form of integral potentials.

Weak solutions of the interior boundary value problems of plane Cosserat elasticity

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.

https://orbilu.uni.lu/bitstream/10993/28980/1/micro-structured-materials-submitted-preprint.pdf

S. Potapenko

Papers

Antiplane shear deformations in a linear theory of elasticity with microstructure

Boundary element analysis of stress distribution around a crack in plane micropolar elasticity

Stress intensity factor for an embedded elliptical crack under arbitrary normal loading

Weak solutions of the interior boundary value problems of plane Cosserat elasticity

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